To determine the largest set on which the function $f(x, y) = \sqrt{x + y} - \sqrt{x - y}$ is continuous, we need to examine the domain of the function. Specifically, the function involves square roots, so we need to ensure that the arguments inside the square roots are non-negative (because square roots are only defined for non-negative real numbers).

Step 1: Conditions for square roots

1. The expression $\sqrt{x + y}$ is defined when: $x + y \geq 0$
2. The expression $\sqrt{x - y}$ is defined when: $x - y \geq 0$

Step 2: Combining the conditions

We need both conditions to be satisfied for $f(x, y)$ to be continuous. Therefore, we need:

1. $x + y \geq 0$
2. $x - y \geq 0$

Let's analyze these two inequalities.

1. The first inequality $x + y \geq 0$ gives: $y \geq -x$

2. The second inequality $x - y \geq 0$ gives: $y \leq x$

Step 3: Describing the region

The inequalities:

• $y \geq -x$
• $y \leq x$

describe a region in the $xy$-plane. The boundary of this region consists of two lines:

1. $y = -x$ (the line with slope -1)
2. $y = x$ (the line with slope 1)

The region where both inequalities are satisfied is the area between these two lines, including the lines themselves.

Step 4: Conclusion

The largest set on which the function $f(x, y) = \sqrt{x + y} - \sqrt{x - y}$ is continuous is the region bounded by and including the lines $y = x$ and $y = -x$. This is the region: $\boxed{ \{ (x, y) \mid -x \leq y \leq x \} }$ or equivalently, $\boxed{ \{ (x, y) \mid |y| \leq x \} }$ for $x \geq 0$.

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Would you like further clarification on any of the steps or a deeper dive into another topic?

Here are some related questions you might find interesting:

1. How does the concept of domain affect the continuity of other functions involving square roots?
2. How do you approach finding the domain of more complex multivariable functions?
3. Can we apply similar methods to determine the continuity of rational functions?
4. What is the relationship between continuity and differentiability in multivariable functions?
5. How do boundary conditions affect the set of points where a function is continuous?

Tip: Always ensure that the arguments of square roots, logarithms, or any other operations with restricted domains are valid (i.e., non-negative for square roots). This helps in identifying the correct domain for the function.